Given $ m \angle RPS = 4x + 27$, $ m \angle QPR = 9x - 115$, and $ m \angle QPS = 107$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Explanation: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {9x - 115} + {4x + 27} = {107}$ Combine like terms: $ 13x - 88 = 107$ Add $88$ to both sides: $ 13x = 195$ Divide both sides by $13$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 9({15}) - 115$ Simplify: $ {m\angle QPR = 135 - 115}$ So ${m\angle QPR = 20}$.